The Christoffel symbols $\Gamma^\mu{}_{\alpha\beta}$ tell you how to take derivatives on a curved manifold. In flat space with Cartesian coordinates, partial derivatives of vectors give you vectors. In curved space (or even flat space with curvilinear coordinates), they don’t — the basis vectors themselves change from point to point. The Christoffel symbols correct for this.

Definition

The Christoffel symbols of the second kind are computed entirely from the metric tensor and its first derivatives:

$$\Gamma^\mu{}{\alpha\beta} = \frac{1}{2} g^{\mu\lambda} \left( \frac{\partial g{\lambda\alpha}}{\partial x^\beta} + \frac{\partial g_{\lambda\beta}}{\partial x^\alpha} - \frac{\partial g_{\alpha\beta}}{\partial x^\lambda} \right)$$

They are symmetric in the lower two indices: $\Gamma^\mu{}{\alpha\beta} = \Gamma^\mu{}{\beta\alpha}$.

In $n$ dimensions, there are $n^2(n+1)/2$ independent Christoffel symbols. In 4D spacetime: 40.

Not a tensor

Despite having three indices, the Christoffel symbols are not a tensor — they don’t transform according to the tensor transformation law. They include an extra inhomogeneous term that depends on the second derivatives of the coordinate transformation. This is why you can always find coordinates where $\Gamma = 0$ at a single point (Riemann normal coordinates), but not globally in curved space.

Example: polar coordinates in flat 2D

Even on a flat plane, non-Cartesian coordinates produce nonzero Christoffel symbols.

In polar coordinates $(r, \theta)$ where $ds^2 = dr^2 + r^2 d\theta^2$:

$$\Gamma^r{}{\theta\theta} = -r \qquad \Gamma^\theta{}{r\theta} = \Gamma^\theta{}_{\theta r} = \frac{1}{r}$$

All others are zero. The symbol $\Gamma^r{}_{\theta\theta} = -r$ is why a particle moving in the $\theta$ direction (constant $r$) appears to “accelerate” inward in polar coordinates — it’s the centripetal term, not real curvature.

Example: sphere

On a sphere of radius $R$ with coordinates $(\theta, \phi)$ where $ds^2 = R^2(d\theta^2 + \sin^2\theta \, d\phi^2)$:

$$\Gamma^\theta{}{\phi\phi} = -\sin\theta\cos\theta \qquad \Gamma^\phi{}{\theta\phi} = \cot\theta$$

These encode how the coordinate grid distorts near the poles.

The covariant derivative

The Christoffel symbols define the covariant derivative $\nabla_\mu$ — the generalisation of the partial derivative that produces tensors from tensors:

$$\nabla_\mu v^\nu = \partial_\mu v^\nu + \Gamma^\nu{}_{\mu\lambda} v^\lambda$$

For a covector: $\nabla_\mu w_\nu = \partial_\mu w_\nu - \Gamma^\lambda{}{\mu\nu} w\lambda$

The covariant derivative is the “correct” derivative on curved spaces. The geodesic equation, the Riemann tensor, and Einstein’s field equations are all built from covariant derivatives — which are built from Christoffel symbols — which are built from the metric.